Risk-premia, carry-trade dynamics, and economic value of currency speculation

Journal of International Money and Finance 31 (2012) 1195–1219

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Journal of International Moneyand Finance

journal homepage: www.elsevier .com/locate/ j imf

Risk-premia, carry-trade dynamics, and economic valueof currency speculation

Christian Wagner*,1

WU Wien – Vienna University of Economics and Business, Austria

JEL classification:F31

Keywords:Exchange ratesUncovered interest parityRisk-premiaCarry-tradeEconomic value

* Institute for Finance, Banking and Insurance,4253; fax: þ43 1 31336 90 4253. .

E-mail address: [email protected] The author is indebted for useful conversatio

referee, Geert Bekaert, Chris D’Souza, Charles EngZechner as well as participants at the meetings of tand seminars at the Vienna University of Economi

0261-5606/$ – see front matter � 2012 Elsevier Ltdoi:10.1016/j.jimonfin.2012.01.013

a b s t r a c t

In this paper, we derive the dynamics and assess the economicvalue of currency speculation by formalizing the concept ofa trader inaction range. We show that exchange rate returnscomprise a time-varying risk-premium and that uncoveredinterest parity (UIP) holds in a speculative sense. The often-cited‘forward bias puzzle’ originates from the omission of the risk-premium in standard UIP tests. Consistent with its popularityamong market professionals, the carry-trade strategy can berationalized as it systematically collects risk-premia, however, theeconomic value generated by bilateral carry-trades is limited.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Tests of foreign exchange market efficiency are typically based on an assessment of uncoveredinterest rate parity (UIP). UIP postulates that the expected change in a bilateral exchange rate is equal tothe forward premium, i.e., given that covered interest rate parity holds, it compensates for the interestrate differential. However, startingwith the seminal work byHansen and Hodrick (1980), Bilson (1981),and Fama (1984), empirical research provides evidence that the forward rate is a biased estimate of thefuture spot rate, finding that the higher interest rate currency tends to not depreciate as much aspredicted by UIP or even appreciates. A consequence of the empirical failure of UIP is that foreign

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ns or constructive comments to Kees Koedijk (the editor), an anonymousel, Alois Geyer, Frank Heinemann, Lucio Sarno, Maik Schmeling, and Josefhe European Finance Association, 2008; German Finance Association, 2008;cs and Business and the Austrian Central Bank.

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C. Wagner / Journal of International Money and Finance 31 (2012) 1195–12191196

exchange excess returns appear to be predictable, i.e. the lagged forward premium has explanatorypower for subsequent excess returns. Attempts to explain this ‘forward bias puzzle’ using, amongothers, risk-premia, consumption-based asset pricing theories, and term-structure models have metwith limited success.

While exchange rate anomalies are usually considered to be macropuzzles, the results of recentresearch suggest to take a look under the microstructure lamppost. Evans and Lyons (2002) argue thatorder-flow conveys information that is relevant to the determination of exchange rates and presentempirical evidence strongly supporting their point. The finding that order-flow drives exchange ratessuggests that scrutinizing the trading behavior of market participants who generate order-flow mayoffer deeper insight into the nature of exchange rate puzzles. Lyons (2001) builds on this idea andargues that the forward bias and the predictability of excess returns might be statistically significantbut nevertheless unimportant in economic terms due to limits to speculation: compared to otherinvestment opportunities, the Sharpe ratios realizable from currency speculation are too small toattract traders’ capital. This presumption that traders allocate capital only if Sharpe ratios exceeda certain threshold implies a range of trader inaction for smaller UIP deviations. Within this range,traders do not produce order-flow aimed at exploiting the forward bias which, as a consequence,remains persistent. This further implies that the slope coefficient in standard UIP regressions, the‘Fama-regressions’, can systematically deviate from its hypothesized theoretical value. Empiricalresearch suggests that bilateral exchange rates are characterized by a statistically persistent buteconomically small forward bias, see Sarno et al., (2006), thus being in-line with the general idea oflimits to speculation.

The present paper formalizes the Lyons (2001) concept of the trader inaction range as a device toassess the economic value of currency speculation. We extend his logic and argue that, both, the slopebut also the intercept in the Fama-regression do not always have to correspond to their standardlyhypothesized values but rather that deviations of one or both might occur as long as these do not allowfor economically significant profits. By economic significancewemean that finding excess returns to bestatistically different from zero is not sufficient in economic terms. Profits can be strictly positive butstill too small to attract capital. We therefore take a two-step approach. First, we formulate speculativependants to the standard UIP test to examine whether currency speculation yields non-zero profits.Second, we judge the economic significance of resulting Sharpe ratios via trader inaction rangesimplied by limits to speculation. The exchange rate dynamics implied by speculative UIP suggest thatexchange rate changes indeed just follow the forward premium but additionally comprise a time-varying risk component which depends on the deviation of the current forward premium from itslong-run mean. We show that the forward bias puzzle reported in previous research stems fromomitting this risk-premium in standard UIP tests. Furthermore, the use of carry-trades aimed atexploiting the forward bias can be rationalized in the presence of such a risk-premium, which isconsistent with the carry-trade’s huge popularity among market participants. We also show that anassessment of economic value within the Fama-regression framework necessitates to take a close lookat the regression constant.While research assessing the statistical significance ofUIP deviations heavilyfocuses on the Fama-regression slope coefficient, disregarding the intercept leads to overestimatingcurrency excess returns and consequently to spurious conclusions with respect to economic value.

Empirically, we find support for speculative UIP and the existence of a risk-premium, the omissionof which results in the forward bias puzzle. Carry-traders are able to collect risk-premia and to generatepositive excess returns, however, the economic value generated by bilateral carry-trades is limited. Theempirical results also support our emphasis to explicitly account for the regression intercept whenjudging the economic value of currency speculation. Overall, our findings are consistent with a limit tospeculation explanation for the persistence of the forward bias. Our results are also in-line with Sarnoet al. (2006) but our analysis goes beyond theirs in that we derive trader inaction ranges analytically.This does not only allow us to empirically test the predictions of the limits to speculation hypothesisbut also to analyze the dynamics of currency speculation and their linkage to risk-premia. A particularlynice feature of our framework is that it can be directly applied to the standard Fama-regression setup.Hence, it equips the large research community working on related studies with an effective testingprocedure which is straight-forward in its implementation and provides material information aboutthe economic relevance of UIP deviations.

C. Wagner / Journal of International Money and Finance 31 (2012) 1195–1219 1197

The remainder of this paper is organized as follows. We briefly review the related literature inSection 2. In Section 3 we derive the speculative pendants to the standard UIP test and describe theexchange rate dynamics implied by speculative UIP. We derive trader inaction ranges to judgeeconomic value in Section 4. We present the empirical results and discuss their implications in Section5. Section 6 offers a conclusion. The appendix describes some technical details. All tables and figuresare gathered at the end of the paper.

2. Related literature on UIP and currency speculation

Uncovered interest parity (UIP) postulates that the expected exchange rate change compensates forthe interest rate differential prevailing for the respective countries. Given that covered interest parityholds, the interest rate differential equals the forward premium. A standard test of UIP is the Fama(1984) regression,

Dstþ1 ¼ aþ bp1t þ εtþ1 (1)

where st denotes the logarithm of the spot exchange rate (domestic price of foreign currency) at time t,pt1 the one-period forward premium, i.e. ft1 � st with ft

1 being the logarithm of the one-period forwardrate, and D a one-period change. The null hypothesis that UIP holds is represented by a being zero andb equalling unity. The common finding that empirical research over the last decades provided andconcentrated on is that b is typically lower than unity and often negative. This indicates that the higherinterest rate currency tends to not depreciate as much as predicted by UIP or even appreciates,apparently allowing for predictable excess returns over UIP. Seminal articles in this area are Hansen andHodrick (1980), Bilson (1981), and Fama (1984), surveys of the literature include Hodrick (1987), Frootand Thaler (1990), Taylor (1995), Lewis (1995), Engel (1996), and Sarno (2005).

Fama (1984) argues that the forward bias may be caused by a time-varying risk-premium that ismore volatile than, and negatively correlated with, the expected rate of currency depreciation.However, traditional risk-based explanations have in general had limited success in explaining theobserved linkages between exchange rates and interest rates; see e.g. Bekaert and Hodrick (1993,2001). In particular, attempts to explain the forward bias puzzle using models of risk-premia suggestthat unrealistically high degrees of risk aversion must be assumed to match the two Fama (1984)conditions; see e.g. Frankel and Engel (1984), Domowitz and Hakkio (1985), Cumby (1988), Mark(1988), Engel (1996). Moreover, it is difficult to explain the rejection of UIP and the forward biaspuzzle by recourse to traditional consumption-based asset pricing theories which allow for departuresfrom time-additive preferences (Backus et al., 1993; Bansal et al., 1995; Bekaert, 1996) and fromexpected utility (Bekaert et al., 1997), or by using popular models of the term-structure of interest ratesadapted to a multi-currency setting (Bansal, 1997; Backus et al., 2001; Brennan and Xia, 2006; Wu,2007; Diez de los Rios, 2009). More recently, risk-based explanations have had success in providingrisk-premium-based explanations for the predictability of excess returns in the cross-section offorward premium-sorted currency portfolios (Lustig et al., 2011) and also in explaining the forward biasin bilateral exchange rates (Sarno et al., 2011). Other recent papers find that currency returns comprisea premium for crash risk (Brunnermeier et al., 2008; Farhi et al., 2009; Jurek, 2009).2,3

While exchange rate anomalies are usually considered to be macro puzzles, the results of recentresearch suggest to take a look under the microstructure lamppost. Evans and Lyons (2002) argue

2 This literature review largely focuses on work that has tried to shed light on the forward bias puzzle by risk-premiumarguments. It should be mentioned that the puzzle could also originate from expectational errors resulting from departuresfrom (strong) rationality, see e.g. Bacchetta et al. (2009). Their finding that predictability in excess returns and predictability inexpectational errors are closely linked, however, does nevertheless not exclude that risk-premia account for the forward biaspuzzle. For instance, the findings of Sarno et al. (2011) that multi-country term-structure models can generate unbiased excessreturn predictions and that these are driven by a set of financial market and macroeconomic variables (not only the forwardpremium) suggest that the predictability link uncovered by Bacchetta et al. (2009) can be explained by limited informationprocessing arguments they refer to in their paper.

3 Another explanation of the puzzle is offered by Verdelhan (2010) based on a model in which investors have preferenceswith external habits.


that order-flow conveys information that is relevant to the determination of exchange rates andpresent empirical evidence strongly supporting their point.4 The finding that order-flow drivesexchange rates suggests that scrutinizing the trading behavior of market participants who generateorder-flow may offer deeper insight into the nature of exchange rate puzzles. Lyons (2001) buildson that idea and suggests a microstructural approach building on institutional realities: Tradersonly allocate capital to currency speculation if they expect a higher Sharpe ratio than from otherinvestment opportunities, i.e. some threshold in terms of the Sharpe ratio has to be exceeded.5

Lyons (2001) argues that returns from currency speculation depend on how far b deviates fromunity. For minor UIP deviations, Sharpe ratios are too small to attract speculative capital, therebyimplying a range of trader inaction in the vicinity of UIP. Lyons (2001) states that bs around �1 or 3are necessary to achieve a Sharpe ratio of 0.4, the long-run performance of a buy-and-hold strategyin US equities. Accordingly, he suggests that a range of b-values between approximately �1 and 3characterizes a trader inaction band, within which b might be statistically different from unity butwithout economic relevance.

The concept of Lyons (2001) is thus related to the argument of Baldwin (1990) that transaction costsimply an inactivity band for uncovered interest speculation. According to his calculations, 1 basis pointof transaction costs may be sufficient to not induce trading on interest rate differentials in the range of1–4 percentage points and exchange rate returns not to offset the interest differential. This line ofreasoning also provides a potential explanation for the findings of Huisman et al. (1998) who show thatUIP holds in periods where absolute forward premia are extremely large. They argue that small forwardpremia in normal times have low forecasting power whereas abnormally large forward premiaaccurately reflect expected exchange rate returns. Normal times would thus coincide with periodswhere traders are inactive whereas there is incentive to trade in abnormal times. Sercu andVandebroek (2005) argue that extreme sampling in the spirit of Huisman et al. (1998) potentiallyreduces noise in forward rates induced by transaction cost and they provide further support for a bandof trader inaction in the presence of transaction costs.

Inspired by the concept of a trader inaction ranged (caused by limits to speculation), Sarno et al.(2006) investigate the relationship between spot and forward rates in a smooth transition regres-sion framework. They report evidence for such a non-linear relationship, allowing for a time-varyingforward bias. The empirical results indicate that UIP does not hold most of the time but (expected)deviations from UIP are economically insignificant, i.e. too small to attract speculative capital. Burnsideet al. (2006) argue that transaction costs and price pressure limit the extent to which traders try toexploit the anomaly. Real world market evidence, however, suggests that the carry-trade strategyaimed at exploiting the forward bias is highly popular among financial institutions and extensivelyused in practice. For instance, Galati and Melvin (2004) and Galati et al. (2007) argue that the use ofcarry-trades is a key driver for the surge of foreign exchange trading activity in recent years. Carry-trades involving the Japanese Yen have been of particular interest during the last years, fora detailed evaluation see e.g. Fong (2010). More generally, Villanueva (2007) provides evidence that theforward premium allows for directional predictability which translates into statistically significantprofits from trading on the forward bias. In this paper, we formalize the concept of the trader inactionrange, which allows us to derive the dynamics of currency speculation, to analyze their linkage to risk-premia, and to assess the economic value attainable.

3. Speculative UIP, risk-premia, and dynamics of speculation

Starting from a static trading approach, i.e. a permanent long (or short) position in the foreigncurrency, which can be viewed as a lower benchmark for speculative efficiency, we motivate

4 Other papers emphasizing the role of order-flow in foreign exchange markets include, among others, Lyons (1995), Ito et al.(1998), Rime (2001), Lyons (2002), Evans and Lyons (2004), Bjønnes and Rime (2005), Dominguez and Panthaki (2006), Evansand Lyons (2006), Taylor and Sager (2008), Rime et al. (2008).

5 Lyons (2001) stresses that speculative capital is allocated based on Sharpe ratios in practice. This empirical reality isimportant for his concept rather than a theoretical rational for why such a behavior arises.


a speculative UIP test on the Fama-regression. Speculative UIP implies that exchange rate dynamicscomprise a time-varying risk-premium and we show that its omission in standard UIP tests causes theforward bias puzzle. We propose a test for this risk-premium and outline the dynamics of excessreturns from the static trading approach as well as the carry-trade.

3.1. Static trading approach: risk-premia and excess return dynamics

Building on the argument of Lyons (2001) that traders use Sharpe ratios to evaluate the perfor-mance of their trading strategies, it is instructive to reparametrize the regression in equation (1) interms of excess returns. We use the standard definition of excess returns given by the differencebetween the exchange rate return and the lagged premium, see e.g. Bilson (1981), Fama (1984), andBackus et al. (1993), Sarno et al. (2006), ERtþ1hDstþ1 � p1t hstþ1 � f 1t , which yields

ERtþ1 ¼ aþ ðb� 1Þp1t þ εtþ1; (2)

where ERtþ1 corresponds to the payoff of a long forward position in the foreign currency entered attime t and maturing at t þ 1. Analogously, �ERtþ1 corresponds to a short position.6 Market effi-ciency arguments suggest that in the long-run excess returns should be zero on average. Given thatthe domestic and the foreign interest rates are stationary, the forward premium reverts toa long-run mean which we denote by mp.7 The long-run average of excess returns, ER, can then bewritten as

ER ¼ aþ ðb� 1Þmp: (3)

Note that, since the Fama-regression is usually estimated by OLS, by the least squares principlethe average residual is zero because the regression includes a constant. The standard procedure toassess whether UIP holds is to test the restrictions a ¼ 0 and b ¼ 1 which implies that ER ¼ 0. If onerelaxes the assumption of risk-neutrality, the average excess return should reflect an unconditionalrisk-premium depending on the riskiness of the foreign currency held. The sign of the premiumdepends on whether holding the foreign currency as compared to the domestic currency isassociated with more risk (e.g. political risk, transition effects) or less (e.g. foreign country is a “safehaven”). For the ease of presentation, we discuss the case where the unconditional risk-premiumis zero below, analogous derivations for a non-zero country risk-premium are provided inAppendix A.

Given that holding the currencies of the two countries is equally risky, the average excess returnshould be zero. Taking a speculative efficiency perspective, one notes that an average excess return ofzero does not only result if a and b exactly correspond to these theoretical values but for any values thatsatisfy the less restrictive relationship a¼�(b� 1)mp. Hence, both parametersmight deviate from theirhypothesized values but still not allow for a non-zero average excess return. In fact, this illustrates thatif one of the parameters deviates from its theoretical value, the other one should do so as well such thatthe average excess return growing with the deviation of the one parameter is reduced by an opposingdeviation of the other one. In our empirical analysis we formally test for the existence of such offsettingeffects which is equivalent to testing whether average profits from the static trading approach are zero.Since previous research usually reports tests on whether b ¼ 1, we formulate our test in terms of b aswell, proposing

Test 1 (Speculative UIP Test): For the parameters of the Fama-regression (1), we test the hypothesisb ¼ 1 � a/mp. If this restriction holds, offsetting effects between a and b exist and average excess returnsfrom the static trading approach are zero.

6 Equivalently, one could enter corresponding spot market and money market transactions.7 We consciously leave interest rate modeling outside the scope of this paper. For the purpose of motivating our arguments it

is sufficient to build on the theoretical and empirical results of previous work that interest rates are mean reverting. We take upthis issue again when we discuss potential extensions to our framework.


For the subsequent derivation we conjecture that the relationship b ¼ 1 � a/mp holds, i.e. weconjecture a minimum level of speculative efficiency. Otherwise, non-zero excess returns could begenerated in the long-run – even though the unconditional country risk-premium is zero – just bytaking a permanent long or short position in the foreign currency. Imposing the restriction on theFama-regression (1) yields

Dstþ1 ¼ a� ap1tmp

þ p1t þ εtþ1 (4)

and rewriting the excess return equation (2) gives

ERtþ1 ¼ a� ap1tmp

þ εtþ1: (5)

The spot rate dynamics as given in equation (4) can be described as follows: the core movement inthe change of the exchange rate corresponds just to the forward premium, pt1, as postulated by UIP.Additionally, Dstþ1 is driven by a constant term, a, and a component,�a(pt1/mp), that is governed by theextent to which the forward premium at t deviates from its long-run mean. Hence, our dynamicssuggest that temporary deviations from UIP are possible, but in the long-run reversion towards theparity condition occurs. Note that this specification is consistent with other exchange rate modelingapproaches established in the literature such as regime switching models, e.g. Engel and Hamilton(1990), vector error correction models, see e.g. Brenner and Kroner (1995), Zivot (2000), and Claridaet al. (2003), and smooth transition regression frameworks recently applied by e.g. Sarno et al.(2006). In this context, a plays a role in determining the reversion to long-run UIP. Defininga ¼ a=mp we can rewrite equations (4) and (5) as

Dstþ1 ¼ a�mp � p1t

�þ p1t þ εtþ1;

ERtþ1 ¼ a�mp � p1t

�þ εtþ1

(6)

where a should be positive, i.e. a should have the same sign as mp, to ensure expedient convergence tolong-run UIP. This, however, suggests that over shorter horizons deviations from UIP occur and thatexcess returns represent a time-varying risk-premium.

Given that the exchange rate process is indeed governed as represented in equation (4), estimatingthe Fama-regression (1) leads to a biased estimate of b due to the omission of �a(pt1/mp):

E½b� ¼ bUIP � a

8><>:cov

hp1t ; p

1t =mp

is2p

9>=>; ¼ 1� a

(1mp

): (7)

As argued above for equation (6), a and mp should have the same sign to ensure a proper reversiontowards long-run UIP. This suggests that the slope coefficient in the Fama-regression will be biaseddownwards from its theoretical UIP value, bUIP ¼ 1, which is consistent with empirical research doc-umenting the forward bias puzzle. Hence, our results contribute to the literature attempting to explainthe puzzle by recourse to risk-premium arguments, for a survey see e.g. Engel (1996), and are in-linewith research suggesting that standard UIP tests may be non-informative in the presence of an omittedrisk-premium, see e.g. Barnhart et al. (1999). Our empirical analysis is based on equation (5) for whichwe present unrestricted estimates as given by

ERtþ1 ¼ a1 þ a2p1tmp

þ εtþ1: (8)

The attempt to explain the forward bias puzzle by recourse to risk-premiumarguments is supported ifa2 is significantlydifferent fromzero and ifone cannot reject thata1¼�a2.Note that the latter is ensured if


onefinds evidence for offsetting effects betweena and b (Test 1);findinga1s�a2wouldbe indicative forthe presence of a non-zero country risk-premium, see Appendix A. Accordingly, we formulate

Test 2 (Risk-Premium Test): For the parameters of regression (8), we test the hypothesesa2 ¼ 0 and a1 ¼ �a2. If the former restriction is rejected and the latter holds, a non-zero risk-premiumexists.

If a non-zero risk-premium exits, the (long-run) dynamics of excess returns from the static tradingapproach can be described by enumerating all possible scenarios. Although we provided a rationaleabove that we expect b < 1 in the long-run, we also present scenarios where b > 1 since we refer tothese scenarios in the next subsection. Overall, the excess return process can be summarized in 12scenarios which we summarize in equation (9). The scenarios 1a, 1b,. 6a, 6b depend on the sign of mp,the relation between pt

1 and mp and the combination of b and a values8:

A “perfect” speculation strategy would take long and short positions such as to always realizepositive excess returns, which requires perfect knowledge or foresight of the long-run forwardpremium mp. This is consistent with the literature showing that the term-structure of forwardpremia, contains useful information for predicting exchange rates; see e.g. Clarida and Taylor(1997), Clarida et al. (2003), and Boudoukh et al. (2006). In practice, however, market partici-pants do not have perfect foresight but can base their investment decisions only on informationavailable at time t.

3.2. Exploiting the forward bias: carry-trade

The empirical evidence that the Fama-b is typically negative has of course also been recognized bypractitioners and motivated the design of trading rules attempting to exploit the forward bias; see e.g.Deutsche Bank (2004). Carry-trade strategies – take a long position in the higher interest rate currency,financed by a short position in the low interest rate currency – are highly popular among marketparticipants; see e.g. Galati and Melvin (2004), Galati et al. (2007).

The excess return from a bilateral carry-trade, CTtþ1, can be written in terms of ERtþ1 introduced inequation (2): conditional on the time-t forward premium, one would sell forward the foreign currencyat time t if pt1 > 0 and realize a payoff of �ERtþ1 at t þ 1; a long position is entered if pt1 < 0, yieldinga payoff of ERtþ1:

CTtþ1 ¼�ERtþ1 ¼ aþ ðb� 1Þp1t þ εtþ1 if p1t < 0;�ERtþ1 ¼ �a� ðb� 1Þp1t � εtþ1 if p1t > 0:

(10)

8 The summary of scenarios is to be read in the following way: for instance, the first line of entries refers to scenarios 1a and1b. Both scenarios refer to situations where mp is positive and the current forward premium is greater than the long-runpremium. In such a situation, the combination of b < 1 and a > 0 results in a negative excess return (scenario 1a) whereasthe combination of b > 1 and a < 0 results in a positive excess return (scenario 1b).


It is instructive to reconcile this representation of carry-trade profits with the excess returndynamics of the static trading approach outlined in the previous section. We summarizing thedynamics of carry-trade profits using the same scenarios as in equation (9) 9:

Although carry-trades are motivated by the intention to profit from b less than unity, positiveexcess returns only emerge in four out of six scenarios where b < 1. While a loss is incurred inscenarios 2a and 5a although b < 1, carry-trades are profitable in scenarios 2b and 5b even thoughb > 1. We discuss the pitfalls of exclusively focusing on b and neglecting offsetting effects of a inSection 4.3. Nevertheless, since we argued in the previous subsection that b should be less than unity,the use of carry-trade strategies can be rationalized as it successfully captures risk-premia in mostscenarios. In particular, it can be viewed as a proxy to the prefect foresight strategy as it conditions onpt1 but not on mp.In order to formulate a test of zero-profitability of carry-trades we rewrite equation (10). Since the

sign of the position taken in the foreign currency is opposite to the sign of the forward premium, i.e.long if pt1<0 respectively short if pt1>0, we adjust the parameters and residuals of the Fama-regressionaccordingly. To indicate that a component i of the regression is adjusted for the position taken, we usesuperscript0, with i0 ¼ sgn [pt1]i. Hence, the excess return from the carry-trade and its unconditionalaverage, CT , can be written as

CTtþ1 ¼ ER0tþ1 ¼ a0 þ ðb� 1Þ�p1t �0þε0tþ1;

CT ¼ a0 þ ðb� 1Þp0 þ ε0:

(12)

Note that, if over the investigated period the sign of the premium changes at least once, a0 is nota constant and the mean of εtþ1

0 is non-zero. Therefore, the means of a0, (pt1)0, and ε0tþ1 are compo-

nents of the average carry-trade excess return CT . Excess returns from the carry-trade are notsignificantly different from zero if the restriction b ¼ 1� ða0 þ ε

0Þ=p0 holds on the parameters inregression (1).

Test 3 (Carry-trade zero profit test): For the parameters of the Fama-regression (1), we test thehypothesis b ¼ 1� ða0 þ ε

0Þ=p0. If this restriction holds, average excess returns from the carry-trade arezero.

3.3. The pitfalls of exclusively focusing on b

The Fama-regression (1) assesses market efficiency as a joint test of rational expectations and risk-neutrality. While rational expectations imply that b ¼ 1 and that the forecast error (εtþ1) is

9 This extended summary of scenarios is to be read in the same way as described for equation (9). Equation (11) additionallyreports the carry-trade profits in the respective scenarios; for instance that carry-trade profits should be positive in scenario 1aand negative in scenario 1b.


uncorrelated with information at t, risk-neutrality suggests that a ¼ 0. A non-zero a would representa constant risk-premium. Hundreds of studies have estimated the Fama-regression for differentexchange rates and sample periods with the focus of discussion always directed towards b. Hodrick(1992) cautioned that interpreting the negative b as evidence that the forward premium mispredictsthe direction of subsequent exchange rate returns may be misleading because authors ignore theconstant even though it is relatively large. Similarly, Bekaert and Hodrick (2009) point out in theirtextbook (Chapter 7, p. 239) that people familiar with the regression result of a negative slopecoefficient

“[.] argue that investors should do the “carry-trade” – that is, borrow in the foreign currency toearn both the higher yield and the expected capital appreciation of the dollar implied by theregression. Unfortunately, this interpretation of the regression is wrong because it ignores thevalue of the constant term.”

Nonetheless, the relevance of a has remained under-researched to date, and we are not aware ofa paper that investigates the role of the constant in more detail or provides an interpretation for theestimates of a.

Our motivation for speculative UIP in Section 3.1 suggests that one should look beyond the questionof whether the slope coefficient equals unity and also consider the intercept. We argued that offsettingeffects between a and b should exist and motivated to test whether b ¼ 1 � a/mp which corresponds tozero-profits from static trading positions in the foreign currency. Given that the hypothesized off-setting effects exist, exclusively focusing on b leads to misestimation of profits generable from staticforeign currency positions: excess returns, ER, will be overestimated (in absolute terms) due toneglecting the offsetting effect by a.

Analogously, the assessment of carry-trade profitability might be spurious if the null of the spec-ulative UIP test holds. If – as expected by carry-traders – b < 1, the following can be said for CT: sincep0 < 0 it follows from b > 1 that ðb� 1Þp0 > 0 but also that a0 < 0, again highlighting the offsettingeffects. Thus, one generates profits from b being lower than unity, but profits are eroded by theconstant, sometimes even leading to a loss despite b < 1 (scenarios 2a and 5a). If b > 1 the reverse istrue, but it is not necessarily the case that one makes a loss even though the strategy is motivated bytrading on a b < 1 (scenarios 2b and 5b). Considering b only, may lead to a spurious appraisal of carry-trade profitability and in particular to an overestimation of profits if b < 1.

In general, disregarding a distorts the assessment of zero-profitability of currency speculation.Consequently, as shown in the next section, also the judgment of economic value based on traderinaction ranges will be distorted.

4. Assessment of economic value by trader inaction ranges

To assess the economic significance of excess returns we derive trader inaction ranges implied bylimits to speculation. First, we directly follow Lyons (2001), subsequently we derive the inaction rangebounds for the static trading approach and the carry-trade. We show for both strategies that dis-regarding a leads to overestimation of excess returns and potentially to spurious conclusions about theeconomic value of currency speculation.

4.1. Inaction range as motivated by Lyons (2001)

In this subsection we derive the trader inaction range following the verbal description of Lyons(2001), suggesting that excess returns and hence Sharpe ratios realizable from UIP deviations solelydepend on b; he does neither consider the effect of a on excess returns nor the impact of b on thestandard deviation of profits. For a given forward premium, Sharpe ratios increase as b deviates fromunity. Traders only allocate speculative capital to currency strategies if Sharpe ratios exceed a certainthreshold (as e.g. given by the long-run performance of a buy-and-hold equity investment), implyingthat b needs to deviate correspondingly far from unity to generate order-flow. This logic suggestsa range of trader inaction for bs close to unity while capital could only be attracted if b over-


respectively undershoots the bounds of this range. In the following, we derive the inaction rangebounds; some technical details are provided in Appendix B.1.

Based on the excess return defined in equation (2), we present the Sharpe ratio and the corre-sponding trader inaction range only considering b but disregarding a, i.e. presuming a¼ 0. However, weaccount for bwhen calculating the standard deviation of ERtþ1. The variance of excess returns is given by

s2ER ¼ ðb� 1Þ2s2p þ s2εþ 2ðb� 1Þcovp;ε (13)

with s denoting the standard deviations and covp,ε the covariance of p and ε. If the Fama-regressionparameters are estimated by OLS, the residuals are orthogonal to the premium by assumption, i.e.covp,ε ¼ 0. Setting a ¼ 0 and combining equations (2) and (13), the Sharpe ratio can be written as

SRER;a ¼ 0 ¼ ðb� 1Þmpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb� 1Þ2s2p þ s2

ε

q : (14)

The numerator changes in proportion to mp as b deviates from unity. However, b also enters thedenominator and the standard deviation increases as b deviates from unity. Thus, for increasingdeviations of b, the Sharpe ratio changes monotonically but only at a decreasing rate, and therefore,from a puremathematical point of view, one could say that speculation is limited since the Sharpe ratiois bounded. It is an empirical matter whether the limiting Sharpe ratios as well as the associated bs areeconomically reasonable.

From equation (14) one can derive the trader inaction range in terms of b, i.e. the bs necessary toachieve a certain Sharpe ratio threshold, SRth, by rearranging and solving the resulting quadraticequation,

b½SRth;a ¼ 0� ¼ �SRthsεffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�m2p � SR2ths

2p

�r þ 1: (15)

The b for which the Sharpe ratio is zero, the center of the inaction range, bc[0, a ¼ 0], is unity andtherefore corresponds to the standardly hypothesized UIP value. Around this center, the upper andlower bound are symmetric, as suggested by Lyons (2001), with the width of the range increasingoverproportionallywith the Sharpe ratio threshold. Note that for very small jmpj extremely large Sharperatio thresholds may be necessary to define the bounds, or put differently, a given SRth might beunreachably high.

4.2. Inaction range for the static trading approach

We now take the impact of a on excess returns explicitly into account as given in equation (2). Sometechnical details are provided in Appendix B.2. The standard deviation can be taken from equation (13)since a as a constant has no impact on the variance. The Sharpe ratio therefore is

SRER ¼ aþ ðb� 1Þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb� 1Þ2s2p þ s2

ε

q : (16)

Compared to presuming a¼ 0, a non-zero a affects the Sharpe ratio by a change proportional to thestandard deviation. Given that offsetting effects between a and (b � 1)mp exist, the Sharpe ratiosimplied by equation (16) will be lower than those from equation (14) where a was set to zero.Furthermore, the Sharpe ratio is not a monotonic function of b anymore; while the Sharpe ratio is stillbounded (with the same limits), the Sharpe ratio does not converge to its extremes with b approachingplus or minus infinity, rather the global optimum occurs when b ¼ ðmpsεÞ=ðaspÞ þ 1.

For a given Sharpe ratio threshold, SRth, the respective b-bounds of the inaction range can be calculatedfrom rearranging equation (16) and solving the resulting quadratic equation. The bounds are given by


b½SRth;a� ¼�amp � SRth

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2s2p þ s2

ε

�m2p � SR2ths

2p

�rm2p � SR2ths

2p

þ 1: (17)

The center of the inaction range, i.e. the b resulting in a Sharpe ratio of zero, corresponds to theb-value hypothesized in the speculative UIP test (Test 1) assessing the profitability of static foreigncurrency positions: bc[0, a] ¼ 1 � a/mp. Hence, for non-zero values of a, the inaction range is notcentered around unity and, furthermore, the bounds are not symmetric around bc[0, a]. There mightalso be situations in which the Sharpe ratio threshold is unreachably high, resulting in the inactionrange to be undefined.

Comparing the bounds derived with a¼ 0 to those derived using the Fama-a, a misinterpretation ofeconomic significance might arise due to the fact that the former differ from the latter in terms of thelevel of the inaction range (different centers) as well with respect to its shape (symmetric vs. asym-metric). Accordingly, we formulate the following prediction.

Prediction 1. Disregarding a leads to overestimation of excess returns and consequently to inaccuratetrader inaction ranges for the static trading approach. If offsetting effects between a and b exist (Test 1), theeconomic value generated by the static trading approach is overstated.

4.3. Inaction range for the carry-trade

The excess return from the carry-trade was presented in equation (12), the corresponding varianceis given by

s2CT ¼ s2a0 þ ðb� 1Þ2s2p0 þ s2ε0 þ 2ðb� 1Þcova0;p0 þ 2cova0;ε0 þ 2ðb� 1Þcovp0;ε0 : (18)

Note that if the sign of the premium changes at least once, a0 is not a constant and therefore alsoaffects the standard deviation of carry-trade returns. Furthermore, the covariances can be differentfrom, although will typically be close to, zero. The Sharpe ratio of the carry-trade is given bySRCT ¼ CT=sCT .

The bounds of the carry-trade inaction range for a given Sharpe ratio threshold can be calculatedfrom rearranging SRCT and solving the following quadratic equation:

ðb� 1Þ2np02 � SR2ths

2p0

oþ ðb� 1Þ

n2�a0p0 þ p0ε0 � SR2th

�cova0;p0 þ covp0;ε0

��oþn

a02 þ ε02 þ 2a0ε0 � SR2th

�s2a0 þ s2

ε0 þ 2cova0;ε0

�o:

(19)

The center of the inaction range is given by bc½0;a� ¼ 1� ða0 þ ε0Þ=p0, corresponding to the value

hypothesized in Test 3 for assessing whether the carry-trade yields non-zero profits. Note that thecenter of the range can be different from unity even if a ¼ 0. Analogously to the inaction range derivedfor the static approach, the bounds can be asymmetric.

Disregarding a by presuming the constant, and thereby also the corresponding covariances, to bezero, again affects the judgement of economic significance. The centers of the respective inactionranges differ by bc½0;a� � bc½0;a ¼ 0� ¼ �a0=p0. If offsetting effects between a and (b � 1)mp exist,one finds that bc[0, a] < bc[0, a ¼ 0] if b < 1 and bc[0, a] > bc[0, a ¼ 0] if b > 1. Given our argumentsand previous empirical evidence that b is typically less than unity, neglecting a potentially results inan inaction range on a too high level and spurious indication of economic significance. Furthermore,the inaction range accounting for a is wider than the range based on a ¼ 0; the magnitude of thiseffect depends on how often a0 changes signs. Based on these arguments we state the followingprediction.


Prediction 2. If b < 1, disregarding a leads to an overestimation of carry-trade profits, underestimation oftheir variance, and consequently to inaccurate trader inaction ranges. If offsetting effects between a andb exist (Test 1), the economic value generated by carry-trades is overstated.

5. Empirical analysis

For our empirical analysis we use monthly spot exchange rates and one-month forward premiaprovided by the Bank for International Settlements. The exchange rates considered are the US Dollarversus the Canadian Dollar (CAD), Swiss Franc (CHF), British Pound (GBP), Japanese Yen (JPY), DanishKrone (DKK), and GermanMark (DEM)which is replaced by the Euro (EUR) from 1999 onwards. For thecombined DEM-EUR series the data covers the period from December 1978 to September 2008, for allother currencies September 1977 to September 2008. As the sample stretches out into the currentfinancial market crisis, we frequently compare full sample results to results based only on data until theend of 2005 in Section 5.1 and we present further robustness checks in Section 5.2.

5.1. Results

5.1.1. Speculative UIP, risk-premia, and dynamics of speculationThe first rows of Table 1 display the results of the Fama-regression (1) as commonly reported in

previous literature. a and b are the parameter estimates with heteroscedasticity and autocorrelationconsistent standard errors in parentheses; standard errors are calculated following Newey and West(1987). All estimates of b are negative, and b ¼ 1 is rejected at the 1 percent level for all currencies.Evidence ismixed for thehypothesisa¼ 0:while it cannot be rejected for CADandDEM-EUR, it is rejectedat the 10 percent level for GBP (p-value is 0.0797), at the 5 percent level for the CHF, and at the 1 percentlevel for the JPY. The joint hypothesis that a¼ 0 and b ¼ 1 is rejected at least at the 5 percent level for allcurrencies. Thus, consistentwithprevious research, standard testsdonot supportUIP. In contrast, applying

Table 1Uncovered interest parity.

CAD CHF GBP JPY DEM-EUR

Fama-regressiona �0.0006 (0.0010) 0.0056 (0.0028) �0.0033 (0.0019) 0.0100 (0.0027) 0.0021 (0.0020)b �0.9858 (0.4728) �1.2915 (0.7288) �2.1322 (1.0570) �2.4954 (0.7205) �1.0907 (0.7979)

Standard UIP testsp[a ¼ 0] [0.5510] [0.0457] [0.0797] [0.0002] [0.2996]p[b � 1] [0.0000] [0.0018] [0.0032] [0.0000] [0.0092]p[a ¼ 0, b � 1] [0.0000] [0.0071] [0.0130] [0.0000] [0.0317]

Test 1 (Speculative UIP)p[b ¼ 1 � a/mp] [0.3947] [0.7118] [0.2867] [0.7632] [0.8341]

Test 2 (Risk-premia)a1 �0.0006 (0.0010) 0.0056 (0.0028) �0.0033 (0.0019) 0.0100 (0.0027) 0.0021 (0.0020)�a2 �0.0013 (0.0003) 0.0063 (0.002) �0.0049 (0.0017) 0.0106 (0.0022) 0.0025 (0.0009)p[a1 ¼ a2] [0.3947] [0.7118] [0.2867] [0.7632] [0.8341]p[a2 ¼ 0] [0.0000] [0.0018] [0.0032] [0.0000] [0.0092]

Test 3 (CT zero-profits)

p

"b ¼ 1� a0 þ ε

0

p0

#[0.0011] [0.4955] [0.0239] [0.2536] [0.0437]

Notes: Results are for09/1977-09/2008 forCAD, CHF,GBP, JPY, and12/1978-09/2008 for thecombinedseriesofDEM(until 12/1998)and EUR (from 01/1999). The table reports the Fama-regression estimates of a and b with heteroscedasticity and autocorrelationconsistent standard errors, following Newey and West (1987), in parentheses. mp denotes the long-run average of the forwardpremium. p[.] denotes the p-value for testing the hypothesis formulated in [.]. The first three p-values are for standard hypothesesappliedwhen testing UIP. Test 1 is the speculativeUIP test thatwe proposed in Section 3.1. Results related to Test 2 are estimates ofregression (8) with standard errors in parantheses and p-values of relevant tests. Test 3 is applied to the Fama-regression (1) andinvestigates whether excess returns from carry-trades are significantly different from zero. Superscript0 indicates that a variable isadjusted for the position taken in the strategy; see (12) in Section 3.2. ε denotes the Fama-regression residual.


Test 1, b¼ 1� a/mp, to assesswhetherUIP holds in a speculative sense, does not reject UIP in a single case.This indicates that the hypothesized offsetting relationship between a and (b� 1)mp exists, implying thataverage excess returns are not significantly different from zero. The offsetting relationship between thetwo components of the average excess return is also illustrated in Fig. 1 by plotting the extent to whicha and (b � 1)mp contribute to average excess returns in a stacked column diagram. The economic impor-tanceofa is indicatedby the absolutemagnitudeof the component in average excess returns ranging from31 to 49 percent. Its relevance is particularly highlighted by the very high t-statistics of testing the nullhypothesis that themeans of the (b� 1)pt1 series are zero. Hence, excess returns appear significantly non-zero when disregarding awhereas accounting for a reveals that they are insignificant.

The existence of these offsetting effects forms the basis for the spot rate process as given in equation(4). The underlying exchange rate dynamics allow for time-varying deviations from UIP and areconsistent with a variety of exchange rate modeling approaches established in the literature; see ourdiscussion in Section 3.1. Our empirical results strongly support that exchange rate changes follow theforward premium in their core but additionally carry a component that depends on the extent to whichthe current forward premium deviates from its long-run mean. All estimates of a2 are significantly

Fig. 1. Composition of average excess returns ðERÞ. Notes: This Figure summarizes the composition of average excess returnsER ¼ a þ (b � 1)mp as in (3). Results are for 09/1977-09/2008 for CAD, CHF, GBP, JPY, and 12/1978-09/2008 for the combined series ofDEM (until 12/1998) and EUR (from 01/1999). The plot shows a stacked column diagram that visualizes the composition of averageexcess returns, ER, based on the values reported in the tabular below the graph. The t-statistics in the last row are for testing the nullhypothesis that the mean of the series of ðb� 1Þp1t is equal to zero.


different from zero at the 1 percent level and the restriction of a1 ¼ �a2 cannot be rejected for anycurrency. Our results show that the standard – yet, to date, rather unsuccessful – argument that theforward bias puzzle reported in the literature is caused by an omitted risk-premium is indeed valid.

Assessing the profitability of carry-trade excess returns as proposed in Test 3, reveals mixedevidence: excess returns are significantly different from zero for CAD, GBP, and DEM-EUR, while not sofor the CHF and JPY.

The existence of offsetting effects between a and b, as supported by the results of Test 1, allows toillustrate the dynamics of excess returns from the static trading approach (ER) and the carry-trade (CT)byenumerating all possible scenarioswhichdependon the sign ofmp, the relation betweenpt

1 andmp and thecombination of b and a values; see Sections 3.1 and 3.2. Since the Fama-b estimates are – consistentwithour priors – below unity for all currencies, only scenarios 1a to 6a are relevant. For the static tradingapproach, Panel A of Table 2 lists thepredicted signs of excess returns for each scenario in thefirst columnand reports the corresponding realizations in the remaining columns. The results show that the excessreturns are signed as predicted. Furthermore, Panel A reports the performance of a static long position inthe foreign currency as well as corresponding results for the perfect foresight strategy i.e. the perfor-mance if one had knowledge about mp and could therefore perfectly predict the next period scenario. Theperformance of the perfect foresight strategy is quite similar across all currencies with Sharpe ratiosranging from0.52 to 0.73. In Panel B, analogous results are reported for the carry-trade. First, we find that

Table 2Dynamics of currency speculation.


Panel A: Static trading approach (ER) and perfect foresight strategy

Predicted sign vs. realized:Scenario 1a � �0.0065 �0.0075 �0.0043Scenario 2a þ 0.0072 0.0056 0.0052Scenario 3a þ 0.0032 0.0112 0.0060Scenario 4a � �0.0016 �0.0084Scenario 5a � �0.0012 �0.0010Scenario 6a þ 0.0035 0.0072

Static long foreign currency position:Mean 0.0007 �0.0007 0.0017 �0.0005 �0.0004SD 0.0163 0.0358 0.0307 0.0341 0.0315SR (p.a.) 0.1524 �0.0687 0.1877 �0.0554 �0.0407

Perfect foresight strategy:Mean 0.0024 0.0063 0.0054 0.0071 0.0049SD 0.0161 0.0352 0.0302 0.0334 0.0312SR (p.a.) 0.5203 0.6147 0.6231 0.7326 0.5413

Panel B: Carry-trade (CT)

Predicted sign vs. realized:Scenario 1a þ 0.0065 0.0075 0.0043Scenario 2a � �0.0072 �0.0056 �0.0052Scenario 3a þ 0.0032 0.0112 0.0060Scenario 4a þ 0.0016 0.0084Scenario 5a � �0.0012 �0.0010Scenario 6a þ 0.0035 0.0072

Carry-trade strategy:Mean 0.0019 0.0016 0.0048 0.0027 0.0038SD 0.0162 0.0357 0.0303 0.0341 0.0313SR (p.a.) 0.4122 0.1573 0.5513 0.2704 0.4165

Notes: Results are for 09/1977-09/2008 for CAD, CHF, GBP, JPY, and 12/1978-09/2008 for the combined series of DEM (until 12/1998) and EUR (from 01/1999). Panel A compares realized excess returns from the static trading approach (ER) to the signspredicted for scenarios 1a–6a; see Section 3.1. Monthly mean and standard deviations as well as annualized Sharpe ratios arereported for a permanent long position in the foreign currency as well as for the prefect foresight strategy. Panel B comparesrealized carry-trade excess returns (CT) to the signs predicted for scenarios 1a–6a; see Section 3.2. Furthermore, monthly meanand standard deviations as well as annualized Sharpe ratios of carry-trades are reported.

Table 3Trader inaction ranges for the static trading approach.


Fama-regressiona �0.0006 (0.0010) 0.0056 (0.0028) �0.0033 (0.0019) 0.0100 (0.0027) 0.0021 (0.0020)b �0.9858 (0.4728) �1.2915 (0.7288) �2.1322 (1.057) �2.4954 (0.7205) �1.0907 (0.7979)Bounds with a ¼ 0bu 4.7784 [1.0000] 2.8733 [1.0000] 3.8047 [1.0000] 2.5900 [1.0000] 4.9571 [1.0000]bc 1.0000 [0.0000] 1.0000 [0.0018] 1.0000 [0.0032] 1.0000 [0.0000] 1.0000 [0.0092]bl �2.7784 [0.9999] �0.8733 [0.2832] �1.8047 [0.3784] �0.5900 [0.0043] �2.9571 [0.9901]Inference with a ¼ 0bu I. I. I. I. I.bc R. R. R. R. R.bl I. I. I. O. I.Bounds with Fama-abu 3.8109 [0.9996] 0.8181 [0.9987] 1.6831 [0.9999] �0.7236 [0.9986] 3.0361 [0.9957]bc 0.1214 [0.3947] �1.0326 [0.7118] �1.0750 [0.2867] �2.3152 [0.7632] �0.7801 [0.8341]bl �3.7723 [0.9675] �2.9844 [0.9888] �3.9836 [0.9630] �3.9872 [0.9919] �4.9705 [0.9875]Inference with Fama-abu I. I. I. I. I.bc N. N. N. N. N.bl I. I. I. I. I.

Notes: Results are for 09/1977-09/2008 for CAD, CHF, GBP, JPY, and 12/1978-09/2008 for the combined series of DEM (until 12/1998) and EUR (from 01/1999). a and b are the estimates of the Fama-regression with heteroscedasticity and autocorrelationconsistent standard errors, following Newey and West (1987), in parentheses. Based on equation (17) and a Sharpe ratiothreshold of 0.5, the upper (bu) and lower (bl) bound aswell as the center (bc) of the inaction range for the static trading approachare calculated, first setting a ¼ 0, second using a from the Fama-regression. The values in square brackets are the p-values fortesting whether b is below bu, b equals bc, and b is above bl. Below the bounds, we summarize these findings by indicatingwhether the hypothesis of b ¼ bc is rejected (R.) or not rejected (N.) and whether b is inside (I.) or outside (O.) the lower boundand the upper bound when comparing the bounds calculated with a ¼ 0 or the Fama-a, respectively.


the realized excess returns are signed aspredicted. Second, theperformance of carry-trades ismixedwithSharpe ratios varying between 0.27 and 0.55. Comparing these figures to the performance of the perfectforesight strategy underpins that the latter dominates and that the carry-trade can be viewed as a simpleproxy for it. Nevertheless, it also shows that carry-trades can be rationalized as they successfully collectrisk-premia. Also note, that the finding that the foresight strategy, which is based on information aboutlong-run interest rates, performs better is consistent with the literature showing that the term-structureof forward premia contains useful information for predicting exchange rates; see e.g. Clarida and Taylor(1997), Clarida et al. (2003), and Boudoukh et al. (2006).

5.1.2. Economic value of bilateral currency speculationTo assess the economic significance of UIP deviations, we report trader inaction ranges for the static

trading approach inTable 3. In afirst step,weuse the full sample up to September 2008 andderive traderinaction ranges based on a Sharpe ratio threshold of 0.5, which Lyons (2001) argues to be reasonablesince the long-run performance of a simple buy-and-hold strategy in US equity is around 0.4.10 The firstrows repeat the Fama-regression estimateswith correspondingNeweyandWest (1987) standard errorsin square brackets. Next, we report the bounds of the trader inaction range when disregarding a, i.e.presuminga¼ 0. budenotes the upper bound, bc the center, and bl the lower bound of the inaction range.The values in parentheses are the p-values for testingwhether b is below the upper bound, whether theestimate is equal to the centerof the range, andwhetherb is above the lower bound.Details of the testingprocedure can be found in Appendix C. The inaction ranges taking a into account are presented in thesame way in the subsequent rows.

The lower and the upper bound derived when presuming a ¼ 0 are symmetrically centered aroundbc ¼ 1, while the bounds derived when using the Fama-a are centered asymmetrically around

10 Lyons (2001), p. 215, states “[.] I feel safe in asserting that there is limited interest at these major institutions in allocatingcapital to strategies with Sharpe ratios below 0.5”.


bc ¼ 1 � a/mp, i.e. the hypothesized value of zero-profits from the static trading approach (Test 1). Notethat the latter bounds do not necessarily even contain the theoretical UIP value of unity. In particular,the results based on the bounds calculated with a ¼ 0 suggest that zero Sharpe ratios are alwaysrejected and even indicate a significant violation of the lower bound for the JPY, pointing at aneconomically significant Sharpe ratio. Incorporating the Fama-a into the assessment reveals that thisfinding is spurious, since for no currency the b is found to be different from the center of the range and,accordingly, bs are always within the inaction range bounds. The finding of whether b is within theinaction range calculated with a ¼ 0 or the Fama-a is summarized in the last three rows by indicatingwhether b ¼ bc is rejected (R.) or not rejected (N.) and whether b is inside (I.) or outside (O.) the lowerbound and the upper bound.

A similar picture evolves when looking at the carry-trade results in Table 4. The inaction rangebounds for a ¼ 0 and the Fama-a respectively differ in level and shape resulting in an inaccurateassessment of economic value if a is disregarded.When setting a¼ 0, zero Sharpe ratios are rejected forall currencies, while this is only the case for CAD, GBP, and DEM-EUR when taking a into account. Withrespect to the lower bound, the results with a ¼ 0 indicate a violation of the lower bound for the JPYthereby suggesting an economically significant Sharpe ratio. Taking account of the Fama-a reveals thatfor none of the currencies b violates the inaction range bounds, again supporting the importance ofconsidering the regression constant when evaluating economic significance.

Our inaction range results for the static trading approach as well as the carry-trade are in favor ofour argument that disregarding amay lead to an inaccurate assessment of economic value. As we onlyfind pronounced evidence for the JPY in both cases, one might at a first glance be tempted to considerthis a JPY-specific phenomenon, though. One has to bear in mind, however, that the sample stretchesuntil September 2008 and hence out into the current crisis period inwhich carry-trade profitability hasbeen greatly reduced. As a consequence, for a given Sharpe ratio threshold of 0.5, economic significancemay not be found independent of whether inaction ranges are calculated accurately or not, but justbecause of the low performance of carry-trades relative to the threshold itself. We therefore reportcarry-trade inaction ranges using only data up to December 2005 in Table 5. The results based on a¼ 0

Table 4Trader inaction ranges for carry-trades.


Fama-regressiona �0.0006 (0.0010) 0.0056 (0.0028) �0.0033 (0.0019) 0.0100 (0.0027) 0.0021 (0.0020)b �0.9858 (0.4728) �1.2915 (0.7288) �2.1322 (1.057) �2.4954 (0.7205) �1.0907 (0.7979)Bounds with a ¼ 0bu 2.5773 [1.0000] 1.9724 [1.0000] 3.4478 [1.0000] 2.3521 [1.0000] 2.8463 [1.0000]bc 0.7122 [0.0004] 0.4256 [0.0190] 1.2738 [0.0014] 0.8646 [0.0000] 0.9152 [0.0124]bl �1.1734 [0.6541] �1.1581 [0.4274] �0.8810 [0.1186] �0.6309 [0.0050] �1.0211 [0.4653]Inference with a ¼ 0bu I. I. I. I. I.bc R. R. R. R. R.bl I. I. I. O. I.Bounds with Fama-abu 2.4302 [1.0000] 0.7582 [0.9974] 2.4381 [1.0000] �0.1690 [0.9993] 2.4568 [1.0000]bc 0.5643 [0.0011] �0.7943 [0.4955] 0.2646 [0.0239] �1.6715 [0.2536] 0.5240 [0.0437]bl �1.3220 [0.7612] �2.4018 [0.9358] �1.9061 [0.4153] �3.2069 [0.8380] �1.4183 [0.6592]Inference with Fama-abu I. I. I. I. I.bc R. N. R. N. R.bl I. I. I. I. I.

Notes: Results are for 09/1977-09/2008 for CAD, CHF, GBP, JPY, and 12/1978-09/2008 for the combined series of DEM (until 12/1998) and EUR (from 01/1999). a and b are the estimates of the Fama-regression with heteroscedasticity and autocorrelationconsistent standard errors, following Newey and West (1987), in parentheses. Based on equation (19) and a Sharpe ratiothreshold of 0.5, the upper (bu) and lower (bl) bound as well as the center (bc) of the inaction range for the carry-trade arecalculated, first setting a¼ 0, second using a from the Fama-regression. The values in square brackets are the p-values for testingwhether b is below bu, b equals bc, and b is above bl. Below the bounds, we summarize these findings by indicating whether thehypothesis of b¼ bc is rejected (R.) or not rejected (N.) and whether b is inside (I.) or outside (O.) the lower bound and the upperbound when comparing the bounds calculated with a ¼ 0 or the Fama-a, respectively.

Table 5Trader inaction ranges for carry-trades until 12/2005.


Fama-regressiona �0.0013 (0.0009) 0.0054 (0.0029) �0.0042 (0.0019) 0.0100 (0.0027) 0.0018 (0.0022)b �1.3714 (0.4144) �1.3316 (0.7338) �2.4778 (1.0639) �2.5434 (0.7374) �1.1993 (0.8051)Bounds with a ¼ 0bu 2.7766 [1.0000] 1.9294 [1.0000] 3.5921 [1.0000] 2.3921 [1.0000] 3.0019 [1.0000]bc 1.0891 [0.0000] 0.3960 [0.0191] 1.4850 [0.0002] 0.8517 [0.0000] 1.0947 [0.0047]bl �0.5920 [0.0304] �1.1762 [0.4162] �0.5892 [0.0384] �0.6975 [0.0064] �0.8068 [0.3131]Inference with a ¼ 0bu I. I. I. I. I.bc R. R. R. R. R.bl O. I. O. O. I.Bounds with Fama-abu 2.3803 [1.0000] 0.8292 [0.9983] 2.2907 [1.0000] �0.0801 [0.9995] 2.6938 [1.0000]bc 0.6902 [0.0000] �0.7100 [0.3975] 0.1841 [0.0128] �1.6359 [0.2193] 0.7857 [0.0142]bl �0.9967 [0.1833] �2.3027 [0.9067] �1.9114 [0.2974] �3.2270 [0.8227] �1.1197 [0.4606]Inference with Fama-abu I. I. I. I. I.bc R. N. R. N. R.bl I. I. I. I. I.

Notes: Results are for 09/1977-12/2005 for CAD, CHF, GBP, JPY, and 12/1978-12/2005 for the combined series of DEM (until12/1998) and EUR (from 01/1999). a and b are the estimates of the Fama-regression with heteroscedasticity and auto-correlation consistent standard errors, following Newey and West (1987), in parentheses. Based on equation (19) anda Sharpe ratio threshold of 0.5, the upper (bu) and lower (bl) bound as well as the center (bc) of the inaction range for thecarry-trade are calculated, first setting a ¼ 0, second using a from the Fama-regression. The values in square brackets arethe p-values for testing whether b is below bu, b equals bc, and b is above bl. Below the bounds, we summarize thesefindings by indicating whether the hypothesis of b ¼ bc is rejected (R.) or not rejected (N.) and whether b is inside (I.) oroutside (O.) the lower bound and the upper bound when comparing the bounds calculated with a ¼ 0 or the Fama-a,respectively.


indicate a violation of the lower bound for three out of the five currencies: CAD, GBP, and JPY. Takingaccount of the Fama-a reveals that for none of the currencies b violates the inaction range bounds andhence – based on the threshold of 0.5 – one would not view bilateral carry-trades as yielding economicvalue.

Our arguments are further strengthened by comparing results across different Sharpe ratiothresholds. While we used the value of 0.5 proposed by Lyons (2001) as an anchor point, theparticular choice of threshold applied for judging economic value depends on a market participant’srisk appetite. We calculate inaction range bounds for the static trading approach and the carry-tradefor a range of Sharpe ratio thresholds for the full sample as well as using only data until the end of2005. Fig. 2 plots the number of currencies for which economic value is indicated for a given Sharperatio threshold depending on whether bounds are calculated based on a ¼ 0 (plotted in gray) or theFama-a (black). For both sample periods, the static trading approach results based on a ¼ 0 indicatethat economic value is generated for at least three currencies up to a Sharpe ratio threshold of 0.3and for one currency (JPY) even beyond 0.5. Accounting for a reveals that none of the Sharpe ratios isdifferent from zero. Analogously, assessing the economic value of carry-trades by inaction rangesbased on a ¼ 0 suggests that Sharpe ratio thresholds of up to 0.8 can be outperformed. Accountingfor a reveals that none of the bilateral carry-trades provides economic value for Sharpe thresholdshigher than 0.4.

The above results provide strong support for our predictions that disregarding a leads to anoverestimation of profits for the static trading approach (Prediction 1) as well as the carry-trade(Prediction 2), hence to inaccurate inaction ranges, and consequently to an incorrect assessmentof economic value.11 Our findings suggest that Sharpe ratios from the static trading approach are

11 As mentioned in Prediction 2, disregarding a when assessing the economic value of carry-trades also leads to an under-estimation of the variance of profits; empirically this effect is relatively small, though.

Fig. 2. Sharpe ratio thresholds and economic significance. Notes: The columns plot the number of currencies for which a givenSharpe ratio threshold (x-axis) is outperformed significantly as judged by trader inaction ranges calculated with a ¼ 0 (columns ingray) and the Fama-a (black) respectively. Overall, five currencies are included: CAD, CHF, GBP, JPY, and the merged series of DEMand EUR. Inaction ranges are calculated for the Static Trading Approach as described in section 3.1 (on the left) and the carry-trade asdescribed in section 3.2 (on the right). The respective data samples are as indicated in the headers of the plots; for the DEM-EUR datais available only from 12/1978.


zero and that carry-trades do not provide economic value for thresholds larger than 0.4. Theseresults are consistent with the Lyons (2001) concept of limits to speculation and in line with theconclusion of Sarno et al. (2006) that the forward bias in bilateral exchange rates is economicallysmall.

Fig. 3. Trader inaction ranges for the carry-trade.Notes: The graphs showthe 60-month rolling Fama-a estimates for the exchange ratesUSD against the indicated foreign currency and the corresponding trader inaction ranges for the carry-trade. The inaction range boundsare calculated with a ¼ 0 (left) and the Fama-a (right) respectively, see equation (19). The underlying Sharpe ratio threshold is 0.5.



5.2. Further robustness checks

With respect to the robustness of our results we examinewhether our conclusions remain the samewhen investigating other currencies, other forward-maturities, or other sample periods. These resultssupport the findings presented above and therefore, to save space, we prefer for most to justsummarize them instead of providing full tables. Detailed results are available upon request.

Apart from the currencies reported in the paper, we have also analyzed a variety of others such asthe Australian Dollar and New Zealand Dollar (which have been excluded because of short dataavailability), other European non-Euro currencies (e.g. Norwegian Krone, Swedish Krone), and furtherEuropean pre-Euro currencies (e.g. French Franc, Italian Lira). The conclusions that can be drawn forthese currencies are qualitatively equivalent to those reached in the paper.

Second, our conclusions are independent of the choice of forward rate maturity. The Bank forInternational Settlements also provides data for three-, six-, and twelve-month horizons.12 Repeatingthe analysis for this data, results are qualitatively the same.

Finally, our findings are robust over time. We did the whole empirical analysis on varioussubsamples and conclusions are qualitatively the same; these results are omitted from the paper tosave space but are available on request. To provide further evidence for the relevance of consideringa in the assessment of economic value, we graph the Fama-b as well as the inaction ranges based ona ¼ 0 and the Fama-a for the carry-trade in Fig. 3. The plots are based on 60-month rolling estimatesusing a Sharpe ratio threshold of 0.5. The graphs illustrate that disregarding a distorts the evaluation ofthe economic significance of speculation profits. While the rolling Fama-bs often seem to undershootthe lower bound when calculating the inaction range with a ¼ 0, this is merely true when accuratelytaking account of the Fama-a. This suggests that our conclusions of limited economic value for bilateralcarry-trades are not particular to the sample period chosen.

6. Conclusion

Tests of foreign exchange market efficiency are typically based on an assessment of uncoveredinterest rate parity (UIP) using the Fama-regression. Empirical research over the last decades consis-tently rejects the UIP condition and claims that the forward rate is a biased estimate of the future spotrate. Attempts to explain this forward bias puzzle, using a variety of (macro-oriented) models, tests,and data, have met with limited success. Recent (microstructure-motivated) research finds that order-flow drives exchange rates which suggests that scrutinizing the trading behavior of market participantsmay offer deeper insight into the nature and the economic relevance of the forward bias puzzle.Traders take up currency speculation strategies, and thereby produce order-flow, only if they expectresulting profits to provide economic value; otherwise they remain inactive and the unexploitedforward bias may remain persistent.

In the present paper, we derived the dynamics and assessed the economic value of currencyspeculation by formalizing the concept of a trader inaction range. We derived a speculative pendant tothe standard UIP condition and showed (i) that exchange rate returns comprise a time-varying risk-premium, (ii) how carry-traders are able to collect this risk-premium, thereby providing a directrationale for the strategy, and (iii) that the forward bias puzzle originates from the omission of the risk-premium in standard UIP tests. Throughout our analysis, we emphasized that focusing on the slopecoefficient but disregarding the intercept in the Fama-regression leads to overestimating excess returnsand consequently to overstating the economic value of currency speculation.

Our empirical results strongly support that UIP holds in a speculative sense and that exchange ratedynamics comprise a time-varying risk-premium, the omission of which causes the forward bias in theFama-regression. Carry-traders generate positive excess returns as predicted which is consistent withmarket evidence that financial institutions routinely apply forward bias strategies. However, the extentto which positive excess returns translate into economic value appears limited.

12 In the context of analyzing different maturities, it is worth mentioning that carry-trades are typically based on (rolling over)short-term contracts since liquidity is higher than for longer maturities.


An appealing feature of our framework is that it can be directly applied to the standard Fama-regression setup. Hence, it equips the large research community working on related studies with aneffective testing procedure which is straight-forward in its implementation and provides materialinformation about the economic relevance of UIP deviations. Since the risk-premium in our frameworkis driven by the current and the long-run forward premium, our results relate to those of (multi-country) term-structure models (e.g. Backus et al., 2001; Brennan and Xia, 2006; Sarno et al., 2011) andour approach could be extended along these lines to account for stochastic interest rates. Anotherextension could be to model a and b as time-varying coefficients which allows for a closer look at therelationship between the two parameters. Moreover, the approach developed in this paper is notlimited in its applicability to foreign exchange markets. The same idea can be extended to othermarkets for which a standard approach is to test expectations hypotheses or unbiasedness hypotheses.

Appendix

Appendix A. Unconditional (country) risk-premium

There might be reasons for the unconditional expectation of ER to be non-zero, in particular, if thecurrencies considered are related to countries with different levels of risk. For instance, one countrymay be considered as a safe heaven as compared to the other, or one country is experiencing a tran-sition. The framework outlined in Section 3.1 does also allow to account for a corresponding uncon-ditional country risk-premium. Let g denote the unconditional expectation of the country risk-premium (which might either be positive or negative), then the average excess return is given by

ER ¼ aþ ðb� 1Þmp ¼ g: (20)

Analogously to Section 3.1, we rearrange this expression in terms of the Fama-regression slopecoefficient

b ¼ 1� a� g

mp(21)

and impose this restriction on the Fama-regression (1). The dynamics of excess returns are then givenby

ERtþ1 ¼ a� ða� gÞ p1t

mpþ εtþ1 (22)

which can again be empirically tested by estimating

ERtþ1 ¼ a1 þ ag2p1tmp

þ εtþ1 (23)

If a2g is significantly different from zero this would again provide evidence for a time-varying riskcomponent as motivated in this paper. However, in the presence of a country risk-premium, one doesnot expect to find that a1 ¼�a2

g. In fact, if one proceeds as described in Section 3.1 and in Test 2, see (8),that a1 s�a2, this is an indication for an unconsidered country risk-premium. Note that the procedurejust described of course also nests a constant risk-premium as discussed in the traditional literaturewhen testing the restriction b ¼ 1 but expecting a s 0. In our framework this would correspond tog ¼ a leading to

Dstþ1 ¼ aþ p1t þ εtþ1 (24)

Appendix B. Sharpe ratios and trader inaction ranges

This appendix summarizes the properties of Sharpe ratios and trader inaction ranges for the statictrading approach. Section B.1 reports technical details when a is disregarded, i.e. presumed to zero,


Section B.2 for calculations based on the Fama-a. We abstain from presenting analogous derivations forthe carry-trade approach, since the details are lengthy but straight-forward along the arguments forthe static trading approach.

B.1. Sharpe ratio and inaction range bounds when a ¼ 0Sharpe Ratio with a ¼ 0: Based on equation (16) we investigate the Sharpe ratio when a ¼ 0,

SR ¼ ðb� 1Þpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb� 1Þ2s2p þ s2

ε

q :

The first derivative of the Sharpe ratio with respect to b is given by

vSRvb

¼ mps2εh

s2εþ ðb� 1Þ2s2p

i3=2;

i.e. depending on the sign of mp, the Sharpe ratio increases (mp > 0) or decreases (mp < 0) monotonically.The second derivative,

v2SR

vb2¼ � 3ðb� 1Þmps2εs2ph


i5=2;shows that, if mp > 0, the Sharpe ratio function is concave (v2SR=vb2 < 0) for b > 1, while it is convex(v2SR=vb2 > 0) for b < 1. The reverse is true if mp < 0. Calculating the limits of the Sharpe ratio functionwith b going to plus and minus infinity reveals that the Sharpe ratio is bounded:

limb/N

SR ¼mp

ffiffiffiffiffis2p

qs2p

and limb/�N

SR ¼ �mp

ffiffiffiffiffis2p

qs2p

:

Inaction Range Bounds with a ¼ 0: Based on equation (15) we investigate the inaction range for UIPdeviations when setting a ¼ 0,

bhSRth;a ¼ 0

i¼ �SRthsεffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

m2p � SR2ths2p

�r þ 1:

To investigate the shape of the inaction range bounded by n upper b, bu and a lower b, bl, we look atthe derivatives with respect to the Sharpe ratio threshold, SRth,

upper bound :vbu

vSRth¼ m2psεh

m2p � s2pSR2th

i3=2 > 0 andv2bu

vSR2th¼ 3m2psεs

2pSRthh

m2p � s2pSR2th

i5=2 > 0;

lower bound :vbl

vSRth¼ � m2psεh

m2p � s2pSR2th

i3=2 < 0 andv2bl

vSR2th¼ � 3m2psεs

2pSRthh

m2p � s2pSR2th

i5=2 < 0:

Thus, the upper bound is an increasing convex function of the Sharpe ratio threshold, while thelower bound is decreasing and concave.


B.2. Sharpe ratio and inaction range bounds when using the Fama-aSharpe ratio with Fama-a: In order to investigate the change in the Sharpe ratio when incorporating

the Fama-a instead of setting a ¼ 0, we look at the partial derivatives:

vSRva

¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2εþ ðb� 1Þ2s2p

q ;

v2SRva2

¼ 0:

Hence, depending on the sign of a, the Sharpe ratio changes inversely proportional to the standarddeviation. Looking at the partial derivatives of the Sharpe ratio with respect to b,

vSRvb

¼ mps2ε� aðb� 1Þs2ph


i3=2;

v2SR

vb2¼ �

3ðb� 1Þmps2εs2p þ as2p

hs2ε� 2ðb� 1Þ2s2p

ihs2εþ ðb� 1Þ2s2p

i5=2 ;

reveals that the function is non-monotonic.While the Sharpe ratio is still boundedwith the same limitsas given above, the global optimum, i.e. vSR=vb ¼ 0, is not reached with b going to plus or minusinfinity but when b ¼ ðmpsεÞ=ðaspÞ þ 1.

Inaction range bounds with Fama-a: To investigate the impact of including a in the assessment ofeconomic significance, we consider the partial derivatives of the inaction range bounds with respectto a:

upperbound :vbu

va¼

�mpþas2pSRthffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2s2pþs2ε

�m2p�s2pSR

2th

�rm2p�s2pSR

2th

andv2bu

va2¼ s2

εs2pSRthh

s2ε

�m2p�s2pSR

2th

�þa2s2p

i3=2>0;

lowerbound :vbl

va¼

�mp�as2pSRthffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2s2pþs2ε

�m2p�s2pSR

2th

�rm2p�s2pSR

2th

andv2bl

va2¼� s2

εs2pSRthh

s2ε

�m2p�s2pSR

2th

�þa2s2p

i3=2<0;

indicating that a non-zero a affects the level as well as the shape of the inaction range.

Appendix C. Testing inaction range bounds

To test whether b significantly overshoots the upper bound or undershoots the lower bound, we usethe non-linear analog to the F statistic; see e.g. Greene (2003) p. 175ff. The general specification

F½J;n� K� ¼ ½Sðb�Þ � SðbÞ�=JSðbÞ=ðn� KÞ (25)

where b* denotes the estimates obtained when the hypothesis is imposed and b denotes the unre-stricted estimates. J is the number of restrictions, n the number of observations, K the number ofparameters. S(.) denotes sum of squared residuals of the estimationwith b* and b respectively. The teststatistic is (approximately) F -distributed with ½J;n� K� degrees of freedom.


One could also use a Wald test which might be simpler to compute. However, as also for the linearcase, the Wald statistic is not invariant to how hypotheses are formulated, potentially leading todifferent answers depending on the specification of the hypothesis. Furthermore, Greene (2003) p. 176states that “the small-sample behavior ofW can be erratic, and themore conservative F-statistic may bepreferable if the sample is not large”.

To judge whether b overshoots the upper bound bu, we want to obtain the probability that b iswithin the inaction range, i.e. whether b < bu. Since our F-test has one numerator degree of freedom,the square-root of the F-statistic corresponds to the absolute value of the t-statistic for the one-sidedtest. Taking the sign of the estimate into account, the probability that b is below bu is therefore given by

the reverse cumulative t-distribution for sgn ½b� bu� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiF½J;n�k�

qwith (n � k) degrees of freedom. If this

probability is below our confidence level threshold, we reject the hypothesis and say that b overshootsthe upper bound. For the lower bound we proceed analogously.

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Risk-premia, carry-trade dynamics, and economic value of currency speculation

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